On indecomposable pure Mendelsohn triple systems

✍ Scribed by F.E. Bennett; Hao Shen

• Publisher: Elsevier Science
• Year: 1991
• Tongue: English
• Weight: 647 KB
• Volume: 97
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(91)90420-7

✦ Synopsis

Bennett,

F.E. and H. Shen, On indecomposable pure Mendelsohn triple systems, Discrete Mathematics 97 (1991) 47-57.

Let u and I be positive integers. A Mendelsohn triple system MTS(u, A) is a pair (X, a), where X is a u-set (of points) and %? is a collection of cyclically ordered 3-subsets of X (called blocks or triples) such that every ordered pair of points of X is contained in exactly A blocks of 58. If we ignore the cyclic order of the blocks, then an MTS(u, J.) can be viewed as a (v, 3, 21)-balanced incomplete block design (BIBD). An MTS(u, A) is called pure if its underlying (u, 3, 2A)-BIBD contains no repeated blocks. An MTS(u, A) is indecomposable if it is not the union of two Mendelsohn triple systems MTS(v, A,) and MTS(u, ,Q with A = A, + h,. For I = 2 and 3, the problem of existence of a pure MTS(u, J.) is completely solved in this paper. We further prove that an indecomposable pure MTS(u, 2) exists if and only if u = 0 or 1 (mod 3) and u 2 6, except possibly u = 9, 12. We show that an indecomposable pure MTS(u, 3) exists for u = 8, 11, 14 and for all u > 17.